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Partial quasimorphisms and quasistates on cotangent bundles, and symplectic homogenization

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  • For a closed connected manifold $N$, we construct a family of functions on the Hamiltonian group $\mathcal{G}$ of the cotangent bundle $T^*N$, and a family of functions on the space of smooth functions with compact support on $T^*N$. These satisfy properties analogous to those of partial quasimorphisms and quasistates of Entov and Polterovich. The families are parametrized by the first real cohomology of $N$. In the case $N=\mathbb{T}^n$ the family of functions on $\mathcal{G}$ coincides with Viterbo's symplectic homogenization operator. These functions have applications to the algebraic and geometric structure of $\mathcal{G}$, to Aubry--Mather theory, to restrictions on Poisson brackets, and to symplectic rigidity.
    Mathematics Subject Classification: Primary: 53D40, 37J50.


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